To describe the theoretical foundations of case-base sampling, we use
the framework of counting processes. In what follows, we abuse notation
slightly and omit any mention of \(\sigma \) -algebras. Instead, following
Aalen et al [1]}, we use the
placeholder “past” to denote the past history of the corresponding
process. The reader interested in more details can refer to Saarela &
Arjas [2]} and Saarela
[3]}. First, let \(N_{i}(t) \in \lbrace 0, 1\rbrace \)  be
counting processes corresponding to the event of interest for individual
\(i=1, \ldots ,n\) . For simplicity, we will consider Type I censoring due
to the end of follow-up at time \(\tau \)  (the general case of
non-informative censoring is treated in Saarela
[3]}). We assume a continuous time model, which
implies that the counting process jumps are less than or equal to one.
We are interested in modeling the hazard functions \(\lambda _{i}(t)\)  of
the processes \(N_i(t)\) , and which satisfy
\(\lambda _{i}(t) dt = E[dN_{i}(t)\mid \mathrm {past}].\) 
